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Natural Logarithms
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
Learning Objective

Recall the basic properties and uses of the natural logarithm
Key Points
 The natural logarithm is the logarithm with base equal to e.
 Also known as Euler's number, e is an irrational number that often appears in natural relationships in pure math and science.
 The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more.
Terms

natural logarithm
The logarithm in base e; either the function that given
$x\!$ returns$y\!$ such that$e^y = x\!$ , or the value of$y\!$ . 
e
The base of the natural logarithm, 2.718281828459045…
Full Text
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The natural logarithm is the logarithm with base equal to e.
The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the yaxis is an asymptote, as seen in .
Interactive Graph: Graph of the Natural Logarithm
Graph of the natural logarithm,
Also known as Euler's number, e is an irrational number representing the limit of:
as n approaches infinity. In other words, e is the sum of 1 plus 1/1 plus 1/(1*2) plus 1/(1*2*3), and so on.
The number e has many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. It also is extremely useful as a base in logarithms; so useful that the logarithm with base e has its own name (natural logarithm) and symbol. Here is the proper notation for the natural logarithm of x:
The natural logarithm is so named because unlike 10, which is given value by culture and has minimal intrinsic use, e is an extremely interesting number that often "naturally" appears, especially in calculus.
The inverse of the natural log appears, for example, upon differentiating a logarithm of any base:
Outside of calculus, the natural logarithm can be used to relate 1, e, i, and π, four of the most important numbers in mathematics:
Key Term Reference
 Interest
 Appears in these related concepts: Accounting for Interest Earned and Principal at Maturity, Interest Compounded Continuously, and Debt Utilization Ratios
 asymptote
 Appears in these related concepts: Standard Equations of Hyperbolas, Basics of Graphing Exponential Functions, and Graphs of Logarithmic Functions
 base
 Appears in these related concepts: Logarithms of Powers, The Role of the Kidneys in AcidBase Balance, and Biology: DNA Structure and Replication
 complex
 Appears in these related concepts: The Elimination Method, The ComplexNumber System, and Electron Transport Chain
 complex numbers
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Phasors, and Complex Numbers
 compound
 Appears in these related concepts: Substances and Mixtures, Compound Inequalities, and Elements and Compounds
 compound interest
 Appears in these related concepts: Calculating Future Value, Calculating Present Value, and MultiPeriod Investment
 constant
 Appears in these related concepts: Inverse Variation, Quadratic Functions of the Form f(x) = a(xh)^2 + k, and Direct Variation
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 exponent
 Appears in these related concepts: Integer Exponents, Logarithms of Products, and Scientific Notation
 function
 Appears in these related concepts: Inverse Functions, Solving Differential Equations, and Modal Mixture
 irrational number
 Appears in these related concepts: Zeroes of Polynomial Functions with Real Coefficients, Domain of a Rational Expression, and Rational Coefficients
 logarithm
 Appears in these related concepts: Solving Problems with Logarithmic Graphs, Changing Logarithmic Bases, and Odds Ratios
 yaxis
 Appears in these related concepts: Quadratic Functions of the Form f(x) = ax^2 + bx + c, Where a is not Equal to 0, Translations, and The Cartesian System
Sources
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Cite This Source
Source: Boundless. “Natural Logarithms.” Boundless Algebra. Boundless, 21 Jul. 2015. Retrieved 23 Jul. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentsandlogarithms5/graphinglogarithmicfunctions37/naturallogarithms1775881/